Dense Output for Strong Stability Preserving Runge–Kutta Methods

Abstract

We investigate dense output formulae (also known as continuous extensions) for strong stability preserving (SSP) Runge–Kutta methods. We require that the dense output formula also possess the SSP property, ideally under the same step-size restriction as the method itself. A general recipe for first-order SSP dense output formulae for SSP methods is given, and second-order dense output formulae for several optimal SSP methods are developed. It is shown that SSP dense output formulae of order three and higher do not exist, and that in any method possessing a second-order SSP dense output, the coefficient matrix A has a zero row.

Appendix: Some Complete Methods in Shu–Osher Form

We have worked with methods in Butcher form because of the simplicity of the order conditions in that form. In this “Appendix” we write out some of the methods with dense output in Shu–Osher form, since this is usually the most convenient form for implementation. For the sake of brevity we write the methods in autonomous form and let ${\mathcal {C}}={\mathcal {C}}(A,b,\overline{{{\varvec{b}}}})$.

The dense output in Shu–Osher form is

$$\begin{aligned} u_{n+\theta }&= \mu u_n + \sum _{j=1}^s {\overline{\beta }}_j(\theta ) \left( y_j + \frac{h}{{\mathcal {C}}}f(y_j)\right) \end{aligned}$$

where

$$\begin{aligned} {\overline{\beta }}^\top&:= \overline{{{\varvec{b}}}}^\top \left( I+{\mathcal {C}}A\right) ^{-1} \\ \mu&:= 1 - {\mathcal {C}}\overline{{{\varvec{b}}}}^\top \left( I + {\mathcal {C}}A\right) ^{-1}. \end{aligned}$$

We denote each method by SSP$(s,p,{\overline{p}})$, where s is the number of stages, p is the order of the method (A, b), and ${\overline{p}}$ is the order of the dense output.

1.1 SSP(2,2,2)

This method has ${\mathcal {C}}(A,b,\overline{{{\varvec{b}}}}) = 1$.

$$\begin{aligned} y_1&= u_n \\ y_2&= y_1 + h f(y_1) \\ u_{n+1}&= \frac{1}{2} u_n + \frac{1}{2} \left( y_2 + hf(y_2)\right) \\ u_{n+\theta }&= \left( 1 - \theta + \frac{1}{2}\theta ^2\right) u_n + (\theta -\theta ^2)\left( y_1+hf(y_1)\right) + \frac{1}{2}\theta ^2\left( y_2 + hf(y_2)\right) . \end{aligned}$$

1.2 SSP(3,2,2)

This method has ${\mathcal {C}}(A,b,\overline{{{\varvec{b}}}}) = 2$.

$$\begin{aligned} y_1&= u_n \\ y_2&= y_1 + \frac{h}{2} f(y_1) \\ y_3&= y_2 + \frac{h}{2} f(y_2) \\ u_{n+1}&= \frac{1}{3} u_n + \frac{2}{3} \left( y_3 + \frac{h}{2} f(y_3)\right) \\ u_{n+\theta }&= \left( 1-2\theta + \frac{4}{3}\theta ^2\right) u_n + 2(\theta -\theta ^2)\left( y_1+\frac{h}{2}f(y_1)\right) + \frac{2}{3}\theta ^2\left( y_3 + \frac{h}{2}f(y_3)\right) . \end{aligned}$$

1.3 SSP(3,3,2)

This method has ${\mathcal {C}}(A,b,\overline{{{\varvec{b}}}}) = 1$.

$$\begin{aligned} y_1&= u_n \\ y_2&= y_1 + h f(y_1) \\ y_3&= \frac{3}{4} u_n + \frac{1}{4} \left( y_2 + h f(y_2)\right) \\ u_{n+1}&= \frac{1}{3} u_n + \frac{2}{3} \left( y_3 + h f(y_3)\right) \\ u_{n+\theta }&= \left( 1 - \theta + \frac{1}{3}\theta ^2\right) u_n + (\theta -\theta ^2)\left( y_1+hf(y_1)\right) + \frac{2}{3}\theta ^2\left( y_3 + hf(y_3)\right) . \end{aligned}$$